1. The PDF's Open in browser only in Internal SharePoint sites. If the PDFs are located in document library for your public-facing website, you will not be able to open them using Internet Explorer.

2. Make sure your Internet Options is not set too high.

3. IF your trying to open the PDF which is attached to the list item while using Internet Explorer, it is not going to work. When using Internet Explorer, the functionality is strictly for SharePoint Online Document Libraries.

4. If you still got issues try using Adobe Reader 10.1.2. UnInstall your current version and Install install Adobe Reader 10.1.2.